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Full text of "Treatise On Analysis Vol-Ii"

5    PRODUCTS OF UNIFORMIZABLE SPACES       17

We have pra(A) <= pra(A) = Aa for all a e I ((12.5.2) and (3.11.4)), hence
A a Y[ Aa. Conversely, if a = (ax) e f| Aa, and if we consider an ele-

a eI                                                             ae I

mentary set U = J| Ua x   fj Ea containing a, its intersection with A is

aeH        ael-H

n (Aanujx n A..

aeH                    ael-H

Since none of the Aa is empty, no Aa is empty and therefore U n A ^ 0,
which shows that ae A.

(12.5.5) Let z \-+f(z) = (/a(r))a e { be a mapping of a topological space F into E.
For f to be continuous at a point ZQ e F, it is necessary and sufficient that each
fa should be continuous at z0.

&

For   any   elementary   set   U = f] Ua x Yl Ea,   we   have

aeH        ael-H

(12.5.6)    Let (jc^n)) be a sequence of points of E. For a = (0a) to be a limit of
the sequence (x(w)), it is necessary and sufficient that aa should be a limit of the
sequence (prax^)n^i9 for each a e I.

This is an immediate consequence of (12.5.5) and the definition of a limit.

In particular, if all the Ea are equal to the same space F, then to say that a
mapping u e F1 of I into F is a limit of a sequence of mappings (un)n^l of I
into F, with respect to the product topology, means that for each a e I the
sequence (t/n(a)) has u(oc) as a limit in F. In this situation we say that the map-
ping u is a simple limit of the sequence (un), or that the latter converges simply
to u.

(12.5.7)    If each space Ea is Hausdorff, then so is the product space E.

For if x 7* y, there exists an index a e I such that pra^c ^ pra<y; hence,
using (12.4.4), there exists X e La such that 4t, *(px<xx> PrajO ^ 0, and there-
fore e^ A(JC, y) 7* 0. The result now follows from (12.4.4).

(12.5.8)    77?e product E of a denumerable jfam/Vy ofmetrizable (resp. separable
metrizable) spaces is metrizable (resp. separable and metrizable).

First of all, E is Hausdorff by (12.5.7). To show that E is metrizable,
apply (12.4.6) and the definition of pseudo-distances on a product. Now

oo

suppose that E = f] Ew, where the En are separable and metrizable. Then each

w=lpen in E.